Haefliger structure

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In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970.[1][2] Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.

Definition[edit]

A codimension- Haefliger structure on a topological space consists of the following data:

  • a cover of by open sets ;
  • a collection of continuous maps ;
  • for every , a diffeomorphism between open neighbourhoods of and with ;

such that the continuous maps from to the sheaf of germs of local diffeomorphisms of satisfy the 1-cocycle condition

for

The cocycle is also called a Haefliger cocycle.

More generally, , piecewise linear, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

Examples and constructions[edit]

Pullbacks[edit]

An advantage of Haefliger structures over foliations is that they are closed under pullbacks. More precisely, given a Haefliger structure on , defined by a Haefliger cocycle , and a continuous map , the pullback Haefliger structure on is defined by the open cover and the cocycle . As particular cases we obtain the following constructions:

  • Given a Haefliger structure on and a subspace , the restriction of the Haefliger structure to is the pullback Haefliger structure with respect to the inclusion
  • Given a Haefliger structure on and another space , the product of the Haefliger structure with is the pullback Haefliger structure with respect to the projection

Foliations[edit]

Recall that a codimension- foliation on a smooth manifold can be specified by a covering of by open sets , together with a submersion from each open set to , such that for each there is a map from to local diffeomorphisms with

whenever is close enough to . The Haefliger cocycle is defined by

germ of at u.

As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are. Indeed, given a continuous map , one can take pullbacks of foliations on provided that is transverse to the foliation, but if is not transverse the pullback can be a Haefliger structure that is not a foliation.

Classifying space[edit]

Two Haefliger structures on are called concordant if they are the restrictions of Haefliger structures on to and .

There is a classifying space for codimension- Haefliger structures which has a universal Haefliger structure on it in the following sense. For any topological space and continuous map from to the pullback of the universal Haefliger structure is a Haefliger structure on . For well-behaved topological spaces this induces a 1:1 correspondence between homotopy classes of maps from to and concordance classes of Haefliger structures.

References[edit]

  • Anosov, D.V. (2001) [1994], "Haefliger structure", Encyclopedia of Mathematics, EMS Press
  1. ^ Haefliger, André (1970). "Feuilletages sur les variétés ouvertes". Topology. 9 (2): 183–194. doi:10.1016/0040-9383(70)90040-6. ISSN 0040-9383. MR 0263104.
  2. ^ Haefliger, André (1971). "Homotopy and integrability". Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School). Lecture Notes in Mathematics, Vol. 197. Vol. 197. Berlin, New York: Springer-Verlag. pp. 133–163. doi:10.1007/BFb0068615. ISBN 978-3-540-05467-2. MR 0285027.